### MATH 5311: Introduction to Geometry and Topology II

**Description:** This course is offered either as an introduction to algebraic topology or differential topology. Differential topology: review of topology; smooth manifolds; vector fields, tangent and cotangent bundles; submersions, immersions and embeddings, submanifolds; Lie groups; differential forms, orientations, integration on manifolds; De Rham cohomolgy, singular cohomology and de Rham theorem; other topics in geometry at the choice of the instructor (e.g. intrinsic Riemannian geometry of surfaces). Algebraic topology: simplical homology, singular homology; Eilenberg-Steenrod axioms; Mayer-Vietoris sequences; CW-homology; cohomology; cup products; Hom and Tensor products; Ext and Tor; the Universal Coefficient Theorems; Cech cohomology and Steenrod homology; Poincare, Alexander, Lefschetz, Alexander-Pontryagin dualities. With a change of content, this course is repeatable to a maximum of six credits.

**Prerequisites:** MATH 5310.

**Credits:** 3

### MATH 5311 - Section 1: Differential Geometry and Topology

**Description:** Topics to be covered:
1. Intersection theory: vector fields, degree, Euler characteristic
2. Integration on manifolds: differential forms, cohomology with forms, Gauss Bonnet theorem
3. Riemannian geometry: curvature, relation between topology and curvature
4. Analysis on manifolds: the Bochner technique, spectral geometry, isoperimetric inequalities. For the topics 1 and 2, I plan to use Differential Topology, by Guillemin and Pollack.
For the topics 3 and 4, I plan to use Riemannian geometry, by Gallot, Hulin and Lafontaine.

**Offered:** Spring

**Credits:** 3

**Sections: **Spring 2015 on Storrs Campus

PSCourseID | Course | Sec | Comp | Time | Room | Instructor |
---|---|---|---|---|---|---|

10722 | 5311 | 001 | Lecture | MWF 10:10:00 AM-11:00:00 AM | MSB311 | Munteanu, Ovidiu |